In the case of the picture you highlight here, while I see how the fading of the photo might give the impression of Confederate grey, the picture is in fact of Cpl. Joseph Pierce of the 14th Connecticut. Born in china, there is some contention on what brought him to America, but he most likely arrived there in 1853 aboard a merchant ship, Hound of Stonington, and settled in the US under the tutelage of her captain, Amos Peck. He enlisted with the regiment in the summer of 1862, was promoted a year later, making him the only known soldier of Chinese heritage to attain NCO rank in the US Army during the war, and served for the duration of the war until the unit disbanded in May, 1865.

Early account of the war claimed Pierce to be one of only a handful of Chinese men serving in the US Army, but research over time has uncovered several dozen. It can be tough to be certain since often there is little more to go off than muster rolls meaning numbers are hard to be completely certain of, but we know of roughly fifty men who served during the American Civil War and were of Chinese heritage. If we expand to count AAPI, the numbers multiply considerably. You can find a list maintained by the National Park Service here which sorts them by country if you want to get a sense of numbers and heritage. It is worth noting that, given a population numbering only in the low hundreds east of the Mississippi, this would suggest that Chinese-Americans enlisted at a rather prodigious rate, their small absolute numbers being a very high percentage of their total population.

Most Chinese-Americans known to have served did so in the American military. Largely this was simply related to the fact that the population was centered in the north, but a few were known to have turned traitor. Chang and Eng Bunker, famous as the original "Siamese twins", had settled in North Carolina where they married local women and settled into life as enslavers, and their children served in rebel grey. Alternatively is the case of John Fouenty, who been kidnapped from Hong Kong and forced to accept a labor contract in Cuba. When completed, he left via a ship that docked in Florida and managed to get himself drafted into the Confederate Army. He promptly deserted at the earliest opportunity however and was allowed to continue on home to China.

If you peruse the NPS list you'll notice that the majority of name identified served in the Navy rather than the Army. In part this reflects the norms of the time, with naval service being generally more open to men of all races to serve, but it also reflects something of a liminal space that those of Asian heritage occupied in the American racial mindset of the time, with census takers of the time in some cases noting Chinese-Ameicans as 'white', and others noting them as 'black' or 'mulatto', the only options available. Many were considered 'white' in the dichotomy of the period, but this doesn't mean they were spared racial prejudice. On both sides we have figures who served in white units, such as Pierce for the US, and the Bunker cousins for the rebels, but also at least two of the Chinese-Americans known to have served did so as members of the US Colored Troops, and it is believed that Yung Wing, who had attended Yale, declined to join because he was not offered a commission as he believed himself entitled to as a college graduate.

After the war race continued to be a point of contention. Almost all Asian-American veterans had been foreign born, and foreign-born (US) veterans were supposed to be guaranteed petition for citizenship. In some cases, Chinese veterans were successful, but others, due to naturalization laws which prohibited Chinese persons from attaining citizenship, were not. The most unfortunate case of this perhaps was William Hang, a US Navy veteran, who successfully applied for citizenship in 1892, only to be suddenly arrested un 1904 for attempting to vote, and despite attempts to appeal, but stripped of citizenship in 1908, being told "I consider the judges who issued the papers responsible for the dense and almost inexcusable ignorance shown by them in not knowing of the naturalization laws [...] which explicitly states that Chinese cannot be naturalized".

Sources

Foenander, Terry et. al.. Asians and Pacific Islanders in the Civil War. National Parks Service, March 2015.

Foenander, Terry et. al.. List of Asian & Pacific Islanders by Country of Origin. National Parks Service

He, Angela (2019) "“Mulatto, Indian, Or What”: The Racialization Of Chinese Soldiers And The American Civil War," The Gettysburg College Journal of the Civil War Era: Vol. 9 , Article 5.

McCunn, Ruthanne Lum. "Chinese in the Civil War: Ten Who Served" Chinese America: History & Perspectives 10: 149-81

McCunn, Ruthanne Lum. "Chinese in the U.S. Civil War" in Chinese Americans: The History and Culture of a People. ed. Jonathan H. X. Lee. ABC-CLIO, 2015.

Page, Charles Davis. History of the Fourteenth Regiment, Connecticut Vol. Infantry. Horton Printing Company, 1906.